Coercive functions from a topological viewpoint and properties of minimizing sets of convex functions appearing in image restoration

Abstract

Many tasks in image processing can be tackled by modeling an appropriate data fidelity term : R^n R \+\ and then solve one of the regularized minimization problems align* &(P_1, ) argmin_x R^n \ (x) \; s.t.\; (x) \ \\ &(P_2, ) argmin_x R^n \ (x) + (x) \, \; > 0 align* with some function : R^n R \+\ and a good choice of the parameter(s). Two tasks arise naturally here: align* & 1. Study the solver sets SOL(P_1,) and SOL(P_2,) of the minimization problems. \\ & 2. Ensure that the minimization problems have solutions. align* This thesis provides contributions to both tasks: Regarding the first task for a more special setting we prove that there are intervals (0,c) and (0,d) such that the setvalued curves align* & SOL(P_1, ), \; (0,c) \\ & SOL(P_2, ), \; (0,d) align* are the same, besides an order reversing parameter change g: (0,c) (0,d). Moreover we show that the solver sets are changing all the time while runs from 0 to c and runs from d to 0. In the presence of lower semicontinuity the second task is done if we have additionally coercivity. We regard lower semicontinuity and coercivity from a topological point of view and develop a new technique for proving lower semicontinuity plus coercivity. Dropping any lower semicontinuity assumption we also prove a theorem on the coercivity of a sum of functions.

Tasks

Reproductions